Anyone who's slightly educated on the Bible knows that it says "judge not, lest ye be judged" (Matthew 7:1). Anyone who's more than slightly educated knows that there is more to the passage. The passage continues to explain the first verse, all the way up to verse 5. Let's read the full passage:
"Judge not, that you be not judged. For with what judgment you judge, you will be judged; and with the measure you use, it will be measured back to you. And why do you look at the speck in your brother’s eye, but do not consider the plank in your own eye? Or how can you say to your brother, ‘Let me remove the speck from your eye’; and look, a plank is in your own eye? Hypocrite! First remove the plank from your own eye, and then you will see clearly to remove the speck from your brother’s eye."
Clearly, Matthew 7:1 is not intended to preclude any form of judgement at all. Rather, Jesus was preaching against hypocritical judgement. The kind of judgement that we do still see in the Church today (and to those Christians I say stop it), but it is not the kind that is typically opposed by those who quote this verse. 9 times out of 10, those who quote Matthew 7:1 without at least alluding to verses 2-5 are sinning. They have done something the Bible says is wrong, they have been directly or indirectly called out on it, and as a dog who has just been told not to eat the cat's food, they're pretty cross about it.
Given that these people are often not wise enough to read more than one verse, they are more than likely unaware of other verses on the topic of judging others. I have for you today four examples. There are other examples I have chosen not to list, partially for brevity's sake, and partially because I would like to encourage you to study for yourselves (2 Timothy 2:15).
The first very important example is John 7:24, which tells us not to judge according to appearance, but to judge righteous judgement. If we are to judge righteous judgement, clearly we are not forbidden to judge at all. In context, Jesus was scolding His persecutors, who sought to kill Him because He had healed a man on the Sabbath (John 5:1-9, 16). The Jews judged Jesus because He appeared to break God's law by healing on the Sabbath, but Jesus was telling them not to judge on the appearance, but to judge righteous judgement. He even backed up His statement by pointing out that they circumcised on the Sabbath, which would also appear to break the law. Therefore, if they judge righteously, they would see that Jesus had done a good thing. For this reason, Christians are also commanded to not judge based on appearance, but judge righteous judgement.
This actually requires a deeper judgement. Judging based on the appearance can be a first moment glance, but judging righteous judgement requires you to actually think through the judgement you are making. And even Jesus submitted to this test, which shows us we can certainly do it to each other,
In Romans 16:17, Christians are urged to note those who cause divisions and preach bad doctrines, avoiding them. This is twofold. First, in order to note those who cause divisions and bad doctrines, we must judge them. There is no logical way around this. Second, it gives us a standard by which to judge. The wording used here is "offenses contrary to the doctrine which you learned", of course meaning that which is now recorded in the scriptures (2 Timothy 3:16-17). The response to these people is to avoid them, meaning we not only have to judge them, but also respond accordingly to that judgement.
Ephesians 5:11 tells us to have no fellowship with the unfruitful works of darkness, but to expose them. This is relevant, because a lot of the time, people use Matthew 7:1 to not only attempt to prevent judgement against a sinning individual, but also against sin in general. Not only do they believe it's wrong to judge the individual, but that it is wrong to say sin is sin. If you are outspoken against abortion, that's seen as being judgemental, even if you're not speaking to, or even around, someone who has a history with abortion. If you are outspoken against the various forms of sexual immorality that are popular in our culture, that's seen as being judgemental too. Pronouncements against an action are seen as being as bad as directly attacking a person. But that's not the case. Christians aren't supposed to be just on the front lines, correcting each other's individual beliefs and behaviours, but we're also supposed to be in the background, showing the entire world the superiority of God's ways over man's. Think of it as like exposing a dodgy fast food restaurant. While popular with those who sit on the tables, you take a look in the kitchen and find it covered in mould and crawling with rats, some of which pee in the coffee machine. When a Christian exposes sin for what it is, they are not sinning, and so citing Matthew 7:1 to try to make them stop is a sin in and of itself.
Titus 1:13 tells us not only to judge, but to strongly rebuke certain people. The liars, evil beasts, and lazy gluttons he speaks of deserve such strong rebuke, and in order to remain sound in the faith, they need it. Thus, not only are we to judge, but we are even capable of using some very strong language in doing so (using wisdom, of course).
Biblically speaking, there are several types of judgement. Some of these are always off limits to us, reserved only for God. Others are permissible, but also off limits to those whose personal sins get in the way. Others are off limits to civilians because it is the job of the government. But there are judgements which the whole Church must be ready to make. Growing in the Christian faith requires judgement. We are to judge deeds and doctrines, groups and individuals and, most importantly, ourselves. Ultimately, anyone who tells you that Christians are never supposed to judge is either uneducated on Christianity, or a grievous liar. I'll leave you to make your own judgements on which they are.